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I came across a problem in a Carmona's Statistical Analysis of Financial Data in R (pg. 189, Problem 3.13). The due date has passed, so now it is considered a self-study question.


I am seeking a simple-to-understand proof to the following:

$X_1 \sim \mathcal{N}(0,1)$, $X_2 \sim \mathcal{N}(0,1)$

$X_3 \sim \begin{cases} |X_2| & X_1 \geq 0 \\ -|X_2| & X_1 < 0 \end{cases} $

(Q.) Compute the CDF of $X_3$. State if $X_3$ is Gaussian.


Intuitively, I believe $X_3$ should be standard normal $\sim \mathcal{N}(0,1)$. Half of the time, $X_3 = $ folded normal, while half of the time $X_3 = $ negative of a folded normal, by simple symmetry. I run an R simulation to test my suspicions and find this to be correct (the R code and plot is listed at the end of the question).

We know $\mathbb{P}(X_1 \leq 0) = \mathbb{P}(X_1 > 0) = \frac{1}{2}$ by symmetry. We also know:

$\mathbb{P}(|X| \leq x) = \\ \mathbb{P}(-x \leq X \leq x) = \\ \mathbb{P}(X \leq x) - \mathbb{P}(X \leq -x) = \\ \mathbb{P}(X \leq x) - [1 - \mathbb{P}(X \leq x)] = \\ 2 \cdot \mathbb{P}(X \leq x) - 1$

So, we try to say $X_3 \sim \frac{1}{2} \cdot [2 \cdot \mathbb{P}(X \leq x) - 1] + \frac{1}{2} \cdot [-2 \cdot \mathbb{P}(X \leq x) + 1]$, but the math here does not work out. Any help from this point would be appreciated.


R Code and Result

# Variables
N <- 1000000
X1 <- rnorm(n=N, mean=0, sd=1); X2 <- rnorm(n=N, mean=0, sd=1)
X3 <- rep(0, N)

# Compute X3
for (i in 1:N) {
  if (X1[i] >= 0) {
    X3[i] <- abs(X2[i])
  } else {
    X3[i] <- -1*abs(X2[i])
  }
}

# Histogram (density)
hist(X3, freq=FALSE, breaks=50, 
     main="Histogram of X3 from Simulation",
     xlab="Value of X3",
     ylab="Density")
# Overlay standard normal for comparison
lines(density(X1), col="blue", lwd=2)

histogram

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  • 2
    $\begingroup$ $X_3$ has a mixture distribution: half the weight is given to a half-normal distribution and the other half of the weight is given to the negative of a half-normal distribution. Together these two halves describe a Normal distribution, QED. In general, the distribution of $X_3$ consists of the positive portion of the distribution of $X_2$, with weight $\Pr(X_1\ge 0)$, along with the reflected version of that positive portion, with the complementary weight. When $X_2$ is symmetric about $0$ and $X_1$ has probability $1/2$ of being non-negative, $X_3$ has the same distribution as $X_2.$ $\endgroup$ – whuber Oct 23 '18 at 2:43
  • $\begingroup$ Crucial assumption of independence of $X_1$ and $X_2$ is missing in the question ( which is certainly mentioned in the original source of the problem). $\endgroup$ – StubbornAtom Oct 25 '18 at 6:12
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I assume $X_1$ and $X_2$ are independent of each other in what follows (thanks for pointing out the importance of this go to @stubbornatom.)

I would work directly with the cumulative density function of $X_3$:

$$P(X_3 \leq x) = 0.5[P(|X_2|\leq x) + P(-|X_2| \leq x)]$$

Consider $x \leq 0$. In this case, $P(|X_2| \leq x) = 0$, so, dropping it from the r.h.s. of the above equation, we have:

$$P(X_3 \leq x) = 0.5P(-|X_2| \leq x) = 0.5[P(X_2 \leq x) + P(-X_2 \leq x)]$$

where the second term on the r.h.s. covers the two cases $X_2 \leq 0$ (in which case $-|X_2| = X_2$) and $X_2 > 0$ (in which case $-|X_2|=-X_2$) respectively. Multiplying the terms of the last probability by $-1$ and switching the direction of the inequality results in:

$$P(X_3 \leq x) = 0.5[P(X_2 \leq x) + P(X_2 \geq -x)] = P(X_2 \leq x)$$

where the last equality is due to the symmetry of the standard Normal distribution around $0$.

A similar logic can be used to show that $P(X_3 \leq x) = P(X_2 \leq x)$ for the case $x > 0$. (Actually I think saying "symmetry" is probably sufficient.) Since the CDF of $X_3$ is everywhere equal to the CDF of $X_2$, and $X_2$ is standard Normal, it follows that $X_3$ is standard Normal too.

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  • $\begingroup$ This is exactly what I was looking for. Thank you for the detailed, step-by-step solution. $\endgroup$ – ERT Oct 23 '18 at 1:14
  • $\begingroup$ Didn't we assume that $X_1$ is independent of $X_2$? $\endgroup$ – StubbornAtom Oct 25 '18 at 6:18
  • $\begingroup$ @StubbornAtom - yes, that's why we can just substitute $0.5$ for $P(X_1 < 0)$ everywhere. $\endgroup$ – jbowman Oct 25 '18 at 14:13
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    $\begingroup$ Was it apparent from the question? That's what I am asking. $\endgroup$ – StubbornAtom Oct 25 '18 at 14:34
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    $\begingroup$ No, it wasn't (+1), but without it you'd be hard-pressed to solve the problem, or so it seems to me. I'll add your point to the top of the anser, because it is important. $\endgroup$ – jbowman Oct 25 '18 at 16:09

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